The difference between $C_p$ and $C_v$ is given by: \begin{equation} C_p-C_v=\left[\left(\frac{\partial U}{\partial V}\right)_T+P\right] \left(\frac{\partial V}{\partial T}\right)_P \end{equation} Substituting into the above equation, $\left(\frac{\partial U}{\partial V}\right)_T =\frac{\alpha}{\kappa}TP$, gives us: \begin{equation} C_p-C_v=\left[\frac{\alpha}{\kappa}T-P+P\right]V\frac {1}{V}\left(\frac{\partial V}{\partial T}\right)_P \end{equation} Simplifying: \begin{equation} C_p-C_v=\frac{TV\alpha^2} {\kappa} \end{equation}